Trigonometrical Equations 3 Question 5
5. Number of solutions of the equation $\tan x+\sec x=2 \cos x$ lying in the interval $[0,2 \pi]$ is
(1993, 1M)
(a) 0
(b) 1
(c) 2
(d) 3
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Answer:
Correct Answer: 5. (c)
Solution:
- $\tan x+\sec x=2 \cos x, x \notin(2 n+1) \frac{\pi}{2}$
$$ \begin{array}{rlrl} \Rightarrow & & \sin x+1 & =2 \cos ^{2} x \\ \Rightarrow & & \sin x+1 & =2\left(1-\sin ^{2} x\right) \\ \Rightarrow & 2 \sin ^{2} x+\sin x-1 & =0 \\ \Rightarrow & (2 \sin x-1)(\sin x+1) & =0 \\ \Rightarrow & & \sin x & =\frac{1}{2}, \sin x=-1 \\ \Rightarrow & & x & =\frac{\pi}{6}, \frac{5 \pi}{6} \\ & & x & =\frac{3 \pi}{2} \\ \text { or } & & x & \notin(2 n+1) \frac{\pi}{2} \\ \text { but } & & x & =\frac{\pi}{6}, \frac{5 \pi}{6} \end{array} $$
Hence, number of solutions are two.
